Precipitates Clusters, Needles, and Platelets

When a moving dislocation line encounters a precipitate that is harder than the matrix in which it is moving, there two ways (in general) for it to get past the precipitate Passing around it or shearing it (i.e., passing through it). 

The loops around the precipitates act as stress concentrators. They exert shearing stresses in addition to the applied stress on the particles. When enough of them have accumulated, the precipitates will be plastically sheared as the loops disappear one by one. This is the basis of a theory of precipitation hardening in an aluminum-copper alloy by Fisher, Hart, and Pry (1953). The precipitate in this case is CuA12, and the precipitates cause an increment of hardening added to the hardness of the solid-solution (Al-Cu) matrix. Quantitative agreement with experimental measurements is fair. 

The line can only behave in this way by being flexible. It has an energy per unit length (line tension). This energy per unit length has two parts  

The elastic strain energy of the material surrounding the core. This energy for a straight line in a large crystal is given approximately by 

U0 = Gb3 where G = elastic shear modulus = 26.2 GPa. for aluminum, so the elastic part of the line energy is about 2.5 eV/atom length, or roughly ten times the core energy. 

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